Problem 1761 (difficulty: 5/10)

Prove the following statements.

(a) If \(\displaystyle T(z)\) is a fractional linear transformation, then \(\displaystyle T\) has a fixed point in \(\displaystyle \CC\cup\infty\).

(b) Given \(\displaystyle z_j\), \(\displaystyle w_j \ \ (j=1, 2, 3)\) with \(\displaystyle (z_k\ne z_j, \ w_k\ne w_j)\), then there is a unique \(\displaystyle T\) fractional linear transformation such that \(\displaystyle T(z_j)=w_j\).

(c) Describe the fractional linear transformations with \(\displaystyle 1\), \(\displaystyle 2\) or more fixed points.

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